Integrand size = 26, antiderivative size = 128 \[ \int \frac {A+B x}{(d+e x)^2 \sqrt {b x+c x^2}} \, dx=\frac {(B d-A e) \sqrt {b x+c x^2}}{d (c d-b e) (d+e x)}-\frac {(b B d-2 A c d+A b e) \text {arctanh}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{2 d^{3/2} (c d-b e)^{3/2}} \]
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Time = 0.06 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {820, 738, 212} \[ \int \frac {A+B x}{(d+e x)^2 \sqrt {b x+c x^2}} \, dx=\frac {\sqrt {b x+c x^2} (B d-A e)}{d (d+e x) (c d-b e)}-\frac {(A b e-2 A c d+b B d) \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{2 d^{3/2} (c d-b e)^{3/2}} \]
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Rule 212
Rule 738
Rule 820
Rubi steps \begin{align*} \text {integral}& = \frac {(B d-A e) \sqrt {b x+c x^2}}{d (c d-b e) (d+e x)}-\frac {(b B d-2 A c d+A b e) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{2 d (c d-b e)} \\ & = \frac {(B d-A e) \sqrt {b x+c x^2}}{d (c d-b e) (d+e x)}+\frac {(b B d-2 A c d+A b e) \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac {-b d-(2 c d-b e) x}{\sqrt {b x+c x^2}}\right )}{d (c d-b e)} \\ & = \frac {(B d-A e) \sqrt {b x+c x^2}}{d (c d-b e) (d+e x)}-\frac {(b B d-2 A c d+A b e) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{2 d^{3/2} (c d-b e)^{3/2}} \\ \end{align*}
Time = 0.53 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.16 \[ \int \frac {A+B x}{(d+e x)^2 \sqrt {b x+c x^2}} \, dx=\frac {\sqrt {x} \left (\frac {\sqrt {d} (B d-A e) \sqrt {x} (b+c x)}{(c d-b e) (d+e x)}-\frac {(b B d-2 A c d+A b e) \sqrt {b+c x} \arctan \left (\frac {-e \sqrt {x} \sqrt {b+c x}+\sqrt {c} (d+e x)}{\sqrt {d} \sqrt {-c d+b e}}\right )}{(-c d+b e)^{3/2}}\right )}{d^{3/2} \sqrt {x (b+c x)}} \]
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Time = 0.52 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.75
method | result | size |
pseudoelliptic | \(\frac {\frac {\left (A e -B d \right ) \sqrt {x \left (c x +b \right )}}{e x +d}-\frac {\left (A b e -2 A c d +B b d \right ) \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )}{\sqrt {d \left (b e -c d \right )}}}{d \left (b e -c d \right )}\) | \(96\) |
default | \(-\frac {B \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {\left (A e -B d \right ) \left (\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )}-\frac {\left (b e -2 c d \right ) e \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{e^{3}}\) | \(370\) |
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Time = 0.41 (sec) , antiderivative size = 399, normalized size of antiderivative = 3.12 \[ \int \frac {A+B x}{(d+e x)^2 \sqrt {b x+c x^2}} \, dx=\left [-\frac {{\left (A b d e + {\left (B b - 2 \, A c\right )} d^{2} + {\left (A b e^{2} + {\left (B b - 2 \, A c\right )} d e\right )} x\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {b d + {\left (2 \, c d - b e\right )} x + 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{e x + d}\right ) - 2 \, {\left (B c d^{3} + A b d e^{2} - {\left (B b + A c\right )} d^{2} e\right )} \sqrt {c x^{2} + b x}}{2 \, {\left (c^{2} d^{5} - 2 \, b c d^{4} e + b^{2} d^{3} e^{2} + {\left (c^{2} d^{4} e - 2 \, b c d^{3} e^{2} + b^{2} d^{2} e^{3}\right )} x\right )}}, -\frac {{\left (A b d e + {\left (B b - 2 \, A c\right )} d^{2} + {\left (A b e^{2} + {\left (B b - 2 \, A c\right )} d e\right )} x\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) - {\left (B c d^{3} + A b d e^{2} - {\left (B b + A c\right )} d^{2} e\right )} \sqrt {c x^{2} + b x}}{c^{2} d^{5} - 2 \, b c d^{4} e + b^{2} d^{3} e^{2} + {\left (c^{2} d^{4} e - 2 \, b c d^{3} e^{2} + b^{2} d^{2} e^{3}\right )} x}\right ] \]
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\[ \int \frac {A+B x}{(d+e x)^2 \sqrt {b x+c x^2}} \, dx=\int \frac {A + B x}{\sqrt {x \left (b + c x\right )} \left (d + e x\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {A+B x}{(d+e x)^2 \sqrt {b x+c x^2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 519 vs. \(2 (112) = 224\).
Time = 0.53 (sec) , antiderivative size = 519, normalized size of antiderivative = 4.05 \[ \int \frac {A+B x}{(d+e x)^2 \sqrt {b x+c x^2}} \, dx=-\frac {\frac {{\left (B b d e^{3} \log \left ({\left | 2 \, c d e - b e^{2} - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c} {\left | e \right |} \right |}\right ) - 2 \, A c d e^{3} \log \left ({\left | 2 \, c d e - b e^{2} - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c} {\left | e \right |} \right |}\right ) + A b e^{4} \log \left ({\left | 2 \, c d e - b e^{2} - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c} {\left | e \right |} \right |}\right ) + 2 \, \sqrt {c d^{2} - b d e} B \sqrt {c} d e {\left | e \right |} - 2 \, \sqrt {c d^{2} - b d e} A \sqrt {c} e^{2} {\left | e \right |}\right )} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{\sqrt {c d^{2} - b d e} c d^{2} {\left | e \right |} - \sqrt {c d^{2} - b d e} b d e {\left | e \right |}} - \frac {2 \, {\left (B d e \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - A e^{2} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )\right )} \sqrt {c - \frac {2 \, c d}{e x + d} + \frac {c d^{2}}{{\left (e x + d\right )}^{2}} + \frac {b e}{e x + d} - \frac {b d e}{{\left (e x + d\right )}^{2}}}}{c d^{2} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{2} \mathrm {sgn}\left (e\right )^{2} - b d e \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{2} \mathrm {sgn}\left (e\right )^{2}} - \frac {{\left (B b d e^{3} - 2 \, A c d e^{3} + A b e^{4}\right )} \log \left ({\left | 2 \, c d e - b e^{2} - 2 \, \sqrt {c d^{2} - b d e} {\left (\sqrt {c - \frac {2 \, c d}{e x + d} + \frac {c d^{2}}{{\left (e x + d\right )}^{2}} + \frac {b e}{e x + d} - \frac {b d e}{{\left (e x + d\right )}^{2}}} + \frac {\sqrt {c d^{2} e^{2} - b d e^{3}}}{{\left (e x + d\right )} e}\right )} {\left | e \right |} \right |}\right )}{{\left (c d^{2} - b d e\right )}^{\frac {3}{2}} {\left | e \right |} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}}{2 \, e^{2}} \]
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Timed out. \[ \int \frac {A+B x}{(d+e x)^2 \sqrt {b x+c x^2}} \, dx=\int \frac {A+B\,x}{\sqrt {c\,x^2+b\,x}\,{\left (d+e\,x\right )}^2} \,d x \]
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